If $f(x)=\log (\sin x), x \in\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right]$, then value of 'c' by applyng LMVT is

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Aicharya · Accepted Answer

Value of 'c' by applying L.M.V.T. is to be found out $\begin{aligned} f(x) &=\log (\sin x) 	ext { on }\left[\frac{\pi}{6}, \frac{5 \pi}{6}ight] \Rightarrow \operatorname{lot} a=\frac{\pi}{6}, b=\frac{5 \pi}{6} \ f^{\prime}(x) &=\frac{\cos x}{\sin x}=\cot x \Rightarrow f^{\prime}(c)=\cot c \ f^{\prime}(c) &=\frac{f(b)-f(a)}{b-a}=\frac{f\left(\frac{5 \pi}{6}ight)-f\left(\frac{\pi}{6}ight)}{5 \frac{\pi}{6}-\frac{\pi}{6}} \ \cot c &=\frac{\log \left[\sin \left(\frac{5 \pi}{6}ight)ight]-\log \left[\sin \left(\frac{\pi}{6}ight)ight]}{\frac{2 \pi}{3}}=\frac{\log \frac{1}{2}-\log \left(\frac{1}{2}ight)}{2 \frac{\pi}{3}} \ \cot c=0 & \Rightarrow c=\frac{\pi}{2} \end{aligned}$

If $f(x)=\log (\sin x), x \in\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right]$, then value of 'c' by applyng LMVT…

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